E6的多项式表示及其组合蕴涵和PDE蕴涵
Polynomial Representation of E6 and Its Combinatorial and PDE Implications
摘要：
本文利用偏微分方程，求出了E6的基本不可约模上的多项式代数分解为不可约子模之和的问题。证明了与Dicksons不变三线性形式对应的三次多项式不变量是唯一的基本不变量。此外，我们还得到了一个组合恒等式，即E6的某些不可约模的维数与26的二项式系数相关。进一步，我们找到了对应于Dickson三线性形式的不变微分算子的所有多项式解，这些多项式解都是关于不可约子模的.
英文摘要：
In this paper, we use partial differential equations to find the decomposition of the polynomial algebra over the basic irreducible module of E6 into a sum of irreducible submodules. It turns out that the cubic polynomial invariant corresponding to the Dicksons' invariant trilinear form is the unique fundamental invariant. Moreover, we obtain a combinatorial identity saying that the dimensions of certain irreducible modules of E6 are correlated by the binomial coefficients of twenty-six. Furthermore, we find all the polynomial solutions for the invariant differential operator corresponding to the Dickson trilinear form in terms of the irreducible submodules.
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